The acronym “SLIC” refers to one approach to addressing the complexities associated with demodulating high-data rate signals, with their generally higher-order modulation constellations. SLIC, or “Serial Localization with Indecision,” uses a multi-stage receiver structure, where the individual stages each operate on a simplified (reduced-order modulation) constellation rather than the full-order modulation constellation(s) associated with the symbols in a received symbol vector.
As noted, SLIC receivers are multi-stage, serial receivers, where each stage includes a joint processing unit. While not limited to Multiple-Input-Multiple-Output (MIMO) scenarios, SLIC represents a particularly advantageous type of reduced-complexity demodulator for MIMO-based receivers. In a MIMO system with N transmit streams, the joint processing unit in a given SLIC stage performs a full search over all candidate N-tuples of centroids. Each centroid represents a subset of symbols from the actual modulation constellation(s) used for transmitting the symbol vector.
The use of centroids creates a residual signal, which belongs to a finite set. The residual signal is effectively modeled as a colored Gaussian noise, having a covariance that is derived from the channel coefficients. Further, the search metric in the joint processing function is the Euclidean metric, with a total covariance that includes that of the residual signal. In a “conventional” SLIC formulation, the joint processing unit at a given SLIC stage produces a unique solution, which is accounted for in the input signals propagated from that given stage to the next stage. Thus at a stage i, the previous (i−1) solutions are unique, and accounted for in its input. With this in mind, it will be appreciated that a key feature of SLIC is the use of overlapping signal subsets. This overlap mitigates the effect of incorrect decisions in early stages on later stages.
Various examples of SLIC receiver structures and associated SLIC processing are known. For example, the interested reader may refer to any one or more of the following published U.S. patent applications: US20110051795A1, US20110051796A1, US20110051851A1, US20110051852A1, US20110051853A1, US20110096873A1, US20110103528A1, US20110243283A1, US20110255638A1, US20110261872A1, and US20120027139A1, all of which are incorporated by reference herein. However, in the interest of presenting helpful background details within this disclosure, consider a conventional SLIC receiver, including a joint processing unit (i.e., a joint demodulator or detector that jointly detects symbols) in each serial stage.
For the MIMO scenario and joint detector, one may consider the N×N case, where there are N transmit antennas and N receive antennas—e.g., a 2×2 MIMO scenarios. For a non-dispersive channel, the received signal isr=Hc+n.  (1)Here r, c and n are N×1 vectors, and H is a N×N matrix. The components of H are independent and Rayleigh faded, and n is white Gaussian noise with covariance R. Further, for this example, assume that all N signals are from the same constellation Q of size q, and all N signals are transmitted with the same power. The effective constellation for c is of size qN.
The optimal receiver in this scenario conducts a full search over all qN candidates c=( c1, . . . , cN)T in QN for one that minimizes the metricD( c|r,R)=(r−H c)HR−1(r−H c),  (2)where subscript T indicates the transpose, and subscript H indicates the Hermitian, or conjugate transpose. The best candidate is denoted ĉ.
The general idea of SLIC is to represent the symbols in a transmitted symbol vector by a series of approximations. In an L-stage SLIC, the symbol vector is effectively represented asc=c[1]+ . . . +c[L],  (3)where stage i detects component c[i], using an effective alphabet Q[i] derived from the true alphabet Q.
While a good general formulation of SLIC-based receiver processing appears in US20110051851A1, as listed above, a few key characteristics are worth expanding herein. To with, in the first stage of a SLIC receiver, the true alphabet Q used for transmission of the symbol vector is approximated by a set of centroids Q[1], of size q[1]<q. Note that Q is also referred to herein as the “actual modulation constellation” for the received symbols of interest.
Each centroid represents a subset of Q. Moreover, the subsets have three properties: (1) the subsets overlap; (2) their union is equal to Q; and (3) all the subsets are shifted versions of the same set O[1] of size o[1], with centroid equal to zero (“0”). The centroid overlap property is a key ingredient, as it enables the indecision feature of SLIC, which in turn boosts demodulation performance.
If L>2, in the second stage O[1] plays the role of Q. That is, O[1] is approximated by a set of centroids Q[2], of size q[2], with the same three properties listed above and based on a set O[2] of size o[2], with centroid 0. We proceed similarly for all stages except the last.
For the last stage L, there is no more approximation, thus Q[L]=O[L−1], and O[L] is empty. The outcome consists of the sets Q[1], . . . , Q[L], which serve as the effective constellations for the L stages of the SLIC receiver.
Consider, for example, the 16-QAM constellation. An example SLIC processing approach “covers” the 16-QAM constellation with four overlapping subsets. This approach is shown in FIG. 1. Each subset is a shifted version of 9-QAM, so that O[1] is 9-QAM. The four respective centroids correspond to 4-QAM or QPSK, so Q[1] is 4-QAM. In a two-stage SLIC receiver, Q[1] serves as the constellation of stage 1, and Q[2]=O[1] as the constellation of stage 2. One may refer to this subset design and to the corresponding SLIC receiver as (4,9).
One may also choose to cover the set O[1] itself with four overlapping subsets, such as shown in FIG. 2. Each subset is a shifted version of 4-QAM, so that O[2] is 4-QAM, as is shown in FIG. 3. The four respective centroids are shown as circles, and also correspond to 4-QAM, so one sets Q[2] to be 4-QAM. In a 3-stage SLIC, Q[1], Q[2] and Q[3]=O[2] serve as the constellations of stages 1, 2 and 3, respectively. One may refer to this subset design and to the corresponding SLIC receiver as (4,4,4).
Note that these subset designs are not unique. For instance, the paper by A. Khayrallah, “SLIC—A Low Complexity Demodulator for MIMO,” Proc. IEEE VTC-Fall, 2011, presents a number of subset designs for 16, 64 and 256-QAM.
Marrying the above details to an example SLIC receiver, FIG. 4 depicts stage i of a known SLIC receiver architecture. The stage input signal is the modified received signal r[i−1] from prior, i.e., the stage (i−1). Of course, for the first stage, the stage input signal is the (starting) received signal of interest. Stage i assumes that the components c[1], . . . , c[i−1] have been determined in earlier stages, and focuses on the demodulation of c[i]. The (stage) residual signalb[i]=(c[i+1]+ . . . +c[L])  (4)belongs to the set O[i]. Stage i models Hb[i], the residual signal filtered by the channel, as a colored Gaussian noise, with covarianceε[i]HHH  (5)where ε[i] is the average energy of the set O[i].
One problematic aspect here is that useful knowledge about O[i] is not being exploited, which is among a number of advantages attending particular embodiments of the present invention. In any case, note that ε[i] decreases as the stage index i grows because there is less and less of the received signal being demodulated that is unaccounted for in the demodulation process as processing progresses through the successive, serial stages of the stages of the SLIC receiver.
The total noise has covarianceR[i]=ε[i]HHH+R  (6)The joint processing unit—i.e., the joint detection unit in each stage—conducts a full search over constellation Q[i]. It searches over all (q[i])N candidates c[i] in (Q[i])N for the one that minimizes the metricD( c[i]|r[i−1],R[i])=(r[i−1]−H c[i])H(R[i])−1(r[i−1]−H c[i])  (7)which is modified from (2) by replacing R with R[i], and the received signal r with the modified received signal r[i−1].
Assuming that stage i is not the last stage, the stage generates as re-modulated signal as {circumflex over (r)}′[i]=Hĉ[i] and then forms a modified received signal for propagation to the next stage as a stage output signal. The modified received signal is expressed asr[i]=r[i−1]−Hĉ[i],  (8)which is fed to the next stage, denoted as stage (i+1). So, each stage i except for the first stage, receives as its stage input signal the modified received signal r[i−1] provided as the stage output signal from the prior stage (i−1). Further, each stage i except for the last stage, outputs its own stage-specific modified received signal r[i], as the stage output signal to be provided to the next stage (i+1) as the r[i−1] stage input signal at that next stage.
As for the first stage where i=1, its stage input signal is the original received signal r. Further, in a non-limiting example of the SLIC architecture, the last stage where i=L, the constellation Q[L] is a subset of Q. Because there is no next stage after the L-th stage, there is no need for a re-modulation block in the L-th stage.
The overall symbol decision made by the SLIC receiver for the received symbol vector is found by adding all the intermediate stage decision vectors ĉ[i] output from each of the SLIC receiver stages. As such, the overall symbol decision is expressed asĉ=ĉ[1]+ . . . +ĉ[L].  (9)The serial arrangement of successive SLIC stages and the overall SLIC structure are shown in FIG. 5.
Note that stage i is affected by the sequence of (i−1) unique previous decisions (ĉ[1], . . . , ĉ[i−1]). This can be seen by writing the input r[i] in (8) explicitly asr[i]=r−H(ĉ[1]+ . . . +ĉ[i−1])=r−Hŝ[i−]  (10)where the decision state of the SLIC receiver at the output of stage i−1 isŝ[i−1]=ĉ[1]+ . . . +ĉ[i−1].  (11)
Thus, ŝ[i−1] summarizes the impact of all preceding stage decisions, i.e., (ĉ[1], . . . , ĉ[i−1]). One may therefore think of ŝ[i−1] as the “state” of the SLIC receiver at the end of processing for stage i−1.
Because of the subset overlap feature, it is possible that a different sequence of candidates ( c[1], . . . , c[i−1]) also satisfies the same value( c[1], . . . , c[i−1])=ŝ[i−1],  (12)meaning that ( c[1], . . . , c[i−1]) would have the same impact as (ĉ[1], . . . , ĉ[i−1]) on stage i. Furthermore, the effect of ( c[1], . . . , c[i−1]) is the same as (ĉ[1], . . . , ĉ[i−1]) on the eventual solution, i.e., the overall symbol decision ĉ in (9). As such, they are interchangeable for purposes of this explanation, thus confirming the notion of ŝ[i−1] as the state of SLIC receiver. Notably, at the input to each stage i, there is only one surviving value of the state ŝ[i−1] provided by the previous stage i−1.